It is an identity. The use of 'i' really not matter. It's just a constant. And it follows right from the Euler's Formula: e^ti = cos t + i sin t which follows from Taylor expansion for e, sin and cos. With t=pi you have e^(pi i) = -1, so ln(-1)/i=pi

But.. We can say that -1 = e^i(pi+2kpi) , can't we..? So, i think that "problem" doesn't invalidate the identity. It's just that we are talking about complex numbers and extending functions can show us some "strange" things if we look at it like we look to the reals. Remember the famous sum: 1+2+3+... = -1/12. It is an identity, but if we look at that as a real sum it makes no sense.

(Sorry if i can't make myself clear enough sometimes. My english is really not good. ;p)